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Theorem recexprlemex 7732

Description: 𝐵 is the reciprocal of 𝐴. Lemma for recexpr 7733. (Contributed by Jim Kingdon, 27-Dec-2019.)

**Hypothesis**
Ref	Expression
recexpr.1	⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩

**Assertion**
Ref	Expression
recexprlemex	⊢ (𝐴 ∈ P → (𝐴 ·_P 𝐵) = 1_P)

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦

**Proof of Theorem recexprlemex**
Step	Hyp	Ref	Expression
1		recexpr.1	. . . 4 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩
2	1	recexprlemss1l 7730	. . 3 ⊢ (𝐴 ∈ P → (1^st ‘(𝐴 ·_P 𝐵)) ⊆ (1^st ‘1_P))
3	1	recexprlem1ssl 7728	. . 3 ⊢ (𝐴 ∈ P → (1^st ‘1_P) ⊆ (1^st ‘(𝐴 ·_P 𝐵)))
4	2, 3	eqssd 3209	. 2 ⊢ (𝐴 ∈ P → (1^st ‘(𝐴 ·_P 𝐵)) = (1^st ‘1_P))
5	1	recexprlemss1u 7731	. . 3 ⊢ (𝐴 ∈ P → (2^nd ‘(𝐴 ·_P 𝐵)) ⊆ (2^nd ‘1_P))
6	1	recexprlem1ssu 7729	. . 3 ⊢ (𝐴 ∈ P → (2^nd ‘1_P) ⊆ (2^nd ‘(𝐴 ·_P 𝐵)))
7	5, 6	eqssd 3209	. 2 ⊢ (𝐴 ∈ P → (2^nd ‘(𝐴 ·_P 𝐵)) = (2^nd ‘1_P))
8	1	recexprlempr 7727	. . . 4 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
9		mulclpr 7667	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
10	8, 9	mpdan 421	. . 3 ⊢ (𝐴 ∈ P → (𝐴 ·_P 𝐵) ∈ P)
11		1pr 7649	. . 3 ⊢ 1_P ∈ P
12		preqlu 7567	. . 3 ⊢ (((𝐴 ·_P 𝐵) ∈ P ∧ 1_P ∈ P) → ((𝐴 ·_P 𝐵) = 1_P ↔ ((1^st ‘(𝐴 ·_P 𝐵)) = (1^st ‘1_P) ∧ (2^nd ‘(𝐴 ·_P 𝐵)) = (2^nd ‘1_P))))
13	10, 11, 12	sylancl 413	. 2 ⊢ (𝐴 ∈ P → ((𝐴 ·_P 𝐵) = 1_P ↔ ((1^st ‘(𝐴 ·_P 𝐵)) = (1^st ‘1_P) ∧ (2^nd ‘(𝐴 ·_P 𝐵)) = (2^nd ‘1_P))))
14	4, 7, 13	mpbir2and 946	1 ⊢ (𝐴 ∈ P → (𝐴 ·_P 𝐵) = 1_P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1372 ∃wex 1514 ∈ wcel 2175 {cab 2190 ⟨cop 3635 class class class wbr 4043 ‘cfv 5268 (class class class)co 5934 1^st c1st 6214 2^nd c2nd 6215 *_Qcrq 7379 <_Q cltq 7380 Pcnp 7386 1_Pc1p 7387 ·_P cmp 7389

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-iinf 4634

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-eprel 4334 df-id 4338 df-po 4341 df-iso 4342 df-iord 4411 df-on 4413 df-suc 4416 df-iom 4637 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-rn 4684 df-res 4685 df-ima 4686 df-iota 5229 df-fun 5270 df-fn 5271 df-f 5272 df-f1 5273 df-fo 5274 df-f1o 5275 df-fv 5276 df-ov 5937 df-oprab 5938 df-mpo 5939 df-1st 6216 df-2nd 6217 df-recs 6381 df-irdg 6446 df-1o 6492 df-2o 6493 df-oadd 6496 df-omul 6497 df-er 6610 df-ec 6612 df-qs 6616 df-ni 7399 df-pli 7400 df-mi 7401 df-lti 7402 df-plpq 7439 df-mpq 7440 df-enq 7442 df-nqqs 7443 df-plqqs 7444 df-mqqs 7445 df-1nqqs 7446 df-rq 7447 df-ltnqqs 7448 df-enq0 7519 df-nq0 7520 df-0nq0 7521 df-plq0 7522 df-mq0 7523 df-inp 7561 df-i1p 7562 df-imp 7564

This theorem is referenced by: recexpr 7733

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